SOLVENT VAPORPRESSURES IN DILUTE SOLUTIONS OF GALLIUM IN CADMIUM
933
Solvent Vapor Pressures in Dilute Solutions of Gallium in Cadmium’v2
by Guy R. B. Elliott, Joe Fred Lemons, and Harold S. Swofford, Jr. University of California Lo8 Ala,mos Scientific Laboratory, Los Alamos, New Mexico (Received September 28, 1964)
Cadmium solvent vapor pressures over solutions a t 775OK. and containing 0.008 to 0.25 mole fraction gallium obey the pressure relationship P C d slloyed/PCd
pure
=
1 - 0.856N~af 0 . 9 4 3 N ~ a ~
The mathematical form of this relationship was deduced by analogy with Henry’s law assuming that solutions are a random mixture of molecules or atoms which interact through short-range forces. The first coefficient, 0.856, was evaluated experimentally from precise and reproducible isopiestic balance measurements. It is presumed to measure the interaction of the gallium solute atoms upon neighboring cadmium solvent atoms. The second coefficient, 0.943 as fitted to the isopiestic measurements of Predel, is included to describe the effect when solvent atoms have two solute neighbors. The data cannot be fitted to the usually accepted modified Raoult’s law expression P C d nlloyed/PCd pure =
1 - N G -I-~ bNGa2
Also a vapor pressure estimate using the Duhem pressure relationship and Kleppa’s calorimetric heats of solution of gallium in cadmium fails to predict the observed cadmium behavior.
There have been surprisingly few adequate experimental tests of the fundamental assumptions of thermodynamics relative to solvent activity. Of interest to this research is Raoult’s law and its modification with a solute-solute interaction term,$ i.e. PA/PAQ =
1 - Ng
+ bNB2
(1)
in which P A o is the vapor pressure of pure solvent; P A is the solvent partial pressure over a solution; N B is the solute mole fraction; b is a constant which measures the effect on the solvent vapor pressure created by the interaction when two solute molecules approach each other sufficiently closely. Using a different approach from that leading to eq. 1 one can arrive at a predicted equation for thc solvent vapor pressure.
+
PAI/PA’ = 1 - ~ A N B
b“B2
(2)
The constant, kA, is designed to measure the effect on the solvent vapor pressure resulting from the interactions of the solute molecules upon their neighboring solvent molecules, and b’ is .introduced to treat the
solvent molecules having two solute neighbors. The development of eq. 2 is presented elsewhere in this paper. Equations 1 and 2 both assume solution randomness and would have to be modified for the amount of order in the solution. Usual experimental practice has been to determine vapor pressures from the midregion of composition out to perhaps 0.9 mole fraction of solvent. Then a b for eq. 1 is evaluated to fit the higher solvent concentration data taken. Such measurements usually do not distinguish adequately between eq. 1 and 2. LA-29972 contains an evaluation of the cases which we found that (1) Work done under the auspices of the U. S.Atomic Energy Commission. (2) Taken in part from Los Alamos Scientific Laboratory Report LA-2997, “An Alternative Treatment of Solvent Activity in t h e Raoult’s Law Region. T h e Gallium-Cadmium System,” by G. R. B. Elliott, J. F. Lemons, and H. S. Swofford, J r . Available for $1.00 from the Office of Technical Services, U. S. Department of Commerce, Washington 25, D. C. (3) Correction for lack of thermodynamic ideality in the vapor phase is here neglected, and it is recognized t h a t a single interaction term is inadequate for an exact representation of the solvent vapor pressure behavior.
Volume 69, Number 3
M a r c h 1965
934
G. R. B. ELLIOTT, J. F. LEMONS,AND H. S. SWOFFORD, JR.
in our opinion test the two relationships; however, the treatment was limited to regions where the bNB2 term was not important. The weight of evidence seemed to us to favor eq. 2 rather strongly, but new experimental evidence was clearly needed. With the development of the isopiestic balance4v6it became possible to measure solvent activities with high precision and apparent high accuracy a t compositions as great as 0.99 mole fraction. Cadmium is convenient as the solvent in the isopiestic balance. The gallium chosen as the solute has several advantages : (1) I t is enough unlike cadmium so that the two liquid elements have a broad range of i nimiscibility6at temperatures sufficiently high so as to eliminate other unusual bonding effects. Thus, a choice between eq. 1 and 2 would not be obscured by an approximation to a perfect solution. (2) The heat of mixing up to 15 atom yo gallium with a cadmium solvent is known to be consistent with Henry’s law modified by a solute-solute. interaction term.’ Therefore the corresponding derived solvent vapor pressure curve has the form of eq. 1 and its constant should be closely predictable through the Duhem pressure relationship coupled with known enthalpies and estimated entropies. (3) Because of the large amounts of solute which must be removed to concentrate a dilute solution, it is inipractical to use a single loading of the isopiestic balance to cover a large range of composition. Therefore, the measurements by Predels could serve both as a check on the isopiestic balance method and as a source of data more useful in evaluating the curvature away from the limiting slope.
Experimental Equipment. The isopiestic balance (Figure 1) provides the means to measure composition and thermodynaniic activil y simultaneously. In this case measurenients were made using volatile cadmium and effectively nonvolatile gallium. A reservoir of pure cadmium at some temperature was allowed to vaporize or condense cadmium so that a gallium-cadmium alloy a t slightly higher temperature could alter its composition and reach pressure equilibrium with the reservoir cadniium. Froin published data the vapor pressure is known as a function of temperature. The temperature difference between the alloy and the reservoir when pressure-coniposition equilibrium has been achieved is thus a measure of the vapor activity over an alloy at that composition and temperature. The isopiestic balance has been described in detail e l s e ~ h e r e . ~Its . ~ construction and operation will be outlined briefly and further discussion will be limited to details pertinent to this series of experiments. The legs The Journal of Physical Chemistry
WEIGHT OF SYSTEM ,-TUNGSTEN SUPPORTED WBY I3 mil R E
S
4
AL HE
ROD
NTER WEIGHT
CADMIUM CONDENSATE HERE
Figure 1. The isopiestic balance (schematic).
of an inverted U-tube of quartz which has been evacuated and sealed provide vapor-connected containers for both the alloy and the cadmium reservoir. This tube is mounted on a quartz frame which is suspended from fine tungsten wires so as to pivot around its approximate center of gravity. When cadmium has transferred between the alloy and the reservoir, the center of gravity has changed. A side arm on the frame is connected by another fine tungsten wire to an analytical balance. Weight at the balance is adjusted to counteract the shift in the center of gravity and thus the change in observed weight is proportional to composition change in the alloy. The balanced quartz tube is surrounded by a furnace consisting of suitable heaters, shields, and insulating brick. The arrangement is such that hotter temperature regions are always positioned higher in the furnace. The layers of insulating bricks are separated by asbestos sheeting, and the outside of the furnace is plastered; thus there is no observable convection problem. Temperature controllers regulate a small fraction of the current flowing to heaters. The heat is applied outside nesting copper cylinders (nickel-plated) which provide both a massive temperature ballast and a means to provide a very uniform temperature around each (4) G. R. B. Elliott and J. F. Lemons, J . Phys. Chem., 64, 137 (1960). (5) G. R. B. Elliott and J. F. Lemons, “Order and Microphases in CeCd-4.6 Solid Solutions,” Advances in Chemistry Series, No. 39, R. F. Gould, Ed., American Chemical Society, Washington, D. C . , 1963, p. 153. (6) T. Heumann and B. Predel, Z. Metallk., 49, 90 (1958). (7) 0 . J. Kleppa, Acta Met., 6, 233 (1958). (8) B. Predel, 2. Metallk., 49, 226 (1958).
SOLVENT VAPORPRESSURES IN DILUTESOLUTIONS OF GALLIUM IN CADMIUM
leg of the inverted U-tube: the heat flows around each cylinder much faster than across the air gap so that temperatures within successive cylinders each become more uniform. Temperatures are measured with l'-Pt-lO% Rh thermocouples positioned about 0.5 in. from the bottoms of the two tube legs. These junctions, hot and cold, are made by cutting both types of wire and forming the two junctions at the cuts. This technique assures that the junctions are of essentially identical composition. The outer copper cylinders are grounded and a grounded stainless steel screen isolates t.he remainder of the thermocouple leads from any external voltage. Either or neither thermocouple lead may be grounded without changing the e.m.f. Three leads out from this couple (center and ends) allow measurement of the temperature difference between the two regions or of the absolute temperature of either region. Temperature differences are measured to hundredths of a degree (0.1 pvolt) and absolute temperature to tenths of a degree. Sample Purzfy. Cadmium of claimed ultrahigh purity was melted and scummed to remove oxide, filed to remove surface material, examined with a microscope for traces of iron filings, and finally vaporized into position to leave any oxide or iron traces behind. Analytical grade gallium was melted and splashed into droplets. Shiny droplets were selected for alloying. To prevent later oxidation of gallium by moisture adsorbed on the quartz, the tube was repeatedly flamed at a temperature near the softening point while being evacuated. The tube was then filled with argon, the metals were introduced, and the tube was re-evacuated. Any remaining moisture would presumably ultimately precipitate gallium out of the alloy as Ga2O3. It seems very unlikely that as much as 5% of the gallium was lost in this way. For reasons of caution we accept this 5% uncertainty, however. The quartz container is not wet by either cadmium or the dilute gallium-in-cadmium alloy. Wetting and direct attack by even unalloyed gallium upon clean quartz under vacuum does not take place unless the temperature is several hundred degrees hotter than the temperatures used for these measurements. The quartz isopiestic balance containers showed no evidence of etching or other attack; by using hydrochloric acid both gallium arid cadmium were readily and completely removed from the containers after the run; the alloy was heated several weeks in the isopiestic balance before measurements were started to assure that any reaction which could take place had done so. Surface Energzes. To evaluate the possibility that surface energies could cause noticeable effect^,^ the
935
bottoms of the tube legs in the second run were made conical while those in the first run were hemispherical cups at the bottom of more narrow tubing. The Measurements. The temperatures for equilibrium between a particular alloy and the reservoir were determined by slowly shifting the reservoir temperature until cadmium stopped transferring into or out of the alloy. The rate of temperature change approaching the balance reversal point was in most cases kept to less than 0.02"K./niin., and this reversal point was approached from both too large and too small teinperatures of the cadmium condensate. Because of the small sample sizes (about 1 g. of alloy initially) and the fact that the system was close to the equilibrium point for hours, it is felt that this constitutes, as far as is experimentally practicable, an essentially equilibrium approach to the point of reversal. Uncertaintzes. Aside from experimental uncertainty in literature values for the vapor pressure of pure cadmium, the principal uncertainties lie in the starting composition (purity, trace oxidation, weight) and in an essentially constant small bias in the temperature difference ( A T ) between the alloy and the pure cadmium reservoir. The bias is created because the cross arm of the reactor tube must be superheated a few degrees to prevent condensation. Some heat from the superheated region flows down the quartz tube legs. 3Iost but not all of this heat is then drawn off by heat sinks near the tube legs. It is this temperature difference which creates an experimentally unavoidable small bias. In the first run the balance factor, i . e . , the relationship of balance shift to composition shift, was established unequivocally and precisely. When the reservoir was slightly hotter than the alloy, all the cadmium was in the alloy and the balance reading was noted. This value was constant, was essentially independent of small changes in temperature, and was reproducible after data points had been measured. When the run was finished and the system was close to temperaturecomposition equilibrium so that the weight changed very slowly, the furnace was partly opened arid the cross arm was quickly collapsed (at the isolator, l'igure 1) using an oxyacetylene flame : Material transfer was stopped before significant cadmium transfer could occur. The system was then cooled and analyzed for how much cadmium was in the reservoir. In this case the relationship between balance reading and composition was determined well enough so that uncertainty in the changes of composition was trivial. In the second run the reservoir temperature control (9) A t the suggestion of Cyril Stanley Smith
Volume 69,S u m b e r 3
March 1866
G. R. B. ELLIOTT, J. F. LEMONS, AND H. S.SWOFFORD, JR.
936
~~
Table I : The Activity of Cadmium Alloyed with Gallium Point no.
1 2 3 4 5 6 7
8 9 10 11
Balance shift
Cd mole fractiona
Alloyb
0.0400 0.3254 1.2095 2 3715 3.0667 0.0924 1.0144 1.6345 3.3660 3 9775 4,4969
0,99191 0.99154 0.99012 0.98733 0.98475 0.99106 0.98941 0,98791 0.98002 0.97403 0.96516
769.89 769.27 771,07 770.48 770.70 774.93 775.20 775.23 775.23 775.16 774,75
-
Temperature, O"K.-Alloy Cadmiumb cross a r m b
769.56 768.92 770.67 769.95 770,07 774.61 774.79 774.75 774,48 774,19 773.50
Cadmium crow armb
785.9 785.6 786.4 785.9 785.7 790.0 790.4 790.2 790.4 789.7 789.4
Cadmium activity'
A TC
785.8 786.3 787.6 786.4 786.0 790.1 789.0 789.3 789,O 787.8 788.5
0 0 0 0 0 0 0 0 0 1 1
33 35 40 53 63 38 44 51 82 04 33
=t0 o l e
0 0 0 0 0 0 0 0 0 0 0
f 0 01 i 0 01' f 0 015 i 0 01 f 0 01 f 0 005 f 0 01 f 0 015 i 0 01 i 0 02
99317 i 0 00021 99274 1 0 00021 99154' 98907 i 0 00032 98703 + 0 00021 99225 i 0 00021 99102 f 0 00011 9 8 9 6 1 5 0 00021 98335 f 0 00032 97875 i 0 00021 97307 f 0 00042
Points 1-5: Initial cadmium 1.04028-g.; gallium 0.00523 g.; 6.18569-g. balance shift = 1.00000-g. cadmium shift. Points 6-11: Initial cadmium 0.95829 g.; gallium 0,00528-g.; 6.2246-g. balance shift = 1.0000-g. cadmium shift. Points 6-11 have less certain balance factor than points 1-5. This leads to some uncertainty a t large cadmium removal from alloy, a s . , points 10 and 11. Measured with Pt-Pt-lO% Rh thermocouple against ice junction to nearest 0.1'K. The greater precision is retained for consistency with AT. iMeasured with Pt-Pt-lO% Rh thermocouple to 0.01"K. temperature difference between alloy and pure cadmium using thermocouple junctions from abutting pieces of thermocouple supply wire so that junction compositions are essentially identical. AT calculated from millivolts using the equation of Roeser and Wenael after subtracting a constant millivolt correction (see Uncertainties) for each run. Points 1-5 reduced by 0.0011 mvolt; points 6-11 reduced by 0.0006 mvolt. For a discussion of usage of the RoeserWenael equation, see ref. 5. Vapor activity relative to that over pure liquid cadmium assuming that Kelley's equation describes the cadmium vapor pressure exactly (see ref. 5). e Uncertainty is one-half the magnitude of the difference between the reversal points approached from cadmium-rich and cadmium-lean alloy compositions. Value approached from only cadmium-rich composition. Activity calculated assuming AT = 0.41.
'
system broke down and caused both uncertainty in one measurement ( N c ~= 0.96516) and an unsuccessful direct analysis of the balance factor. For this reason the balance factor had to be determined by secondary means. First, the various lower arms were measured and a balance factor calculated. Second, a tube of the same diniensions was loaded with only cadmium and a balance factor was determined by the balance readings with the cadmium all on one side or all on the other. The first and second evaluations agreed within experimental error. Finally, this balance factor gave results which agree with the earlier run. There is uncertainty due to the balance factor in this run as is indicated Figure 2. It is significant only where it applies to large removals of cadmium from the known original alloy composition.
1
The Journal of Physical Chemistry
1.00
0.70 -
0.60 -
/ 9'.4'
Results The data for the observed composition of the solution as a function of cadniium pressure are shown in Table I and plotted in Figure 2. Certain comments are pertinent to an evaluation of the data: (1) The total dissolved gallium in each run is constant but. as discussed under Sample Purity, may be uncertain by perhaps 5%. (2) In run 1 each variation in the cadniium content is known to about 1 part in
'
090
ABOVE DATA FROM PREOEL AT 801 *K
050 0
0.w
CIRCLE CIRCLE INMCATES INMCATES ACTIVITY ACTIVITY
UNCERTAINTY UNCERTAINTY. BAR ADOED WHERE COMPOSITIONUNCERTAINTY BECOMES IMPOR-
/';/ asm
TANl
I
0975
oso
1
0985
I a990
I
aws
i
i
a975
10 9 7 0
LOO
CADMIUM LIQUID MOLE FRACTION
Figure 2. Vapor activities over gallium-cadmium solutions.
30,000; for run 2 the uncertaint'ies are indicated in the figure. (3) A temperature bias correction (see Uncer-
SOLVENT VAPORPRESSURES IX DILUTESOLUTIOKS OF GALLIUM IN CADMIUM
tainties) has been applied to each of the measurements of the temperature difference, AT; in run 1 the correction is 0.12OK.; in run 2, 0.07OK. The larger correction corresponds to about 1 part in 500 in the absolute vapor pressure or 3 cal. in the cadmiuni partial molal free energy. (4) The comparative temperature difference between the alloy and the pure cadmium, A(AT), is reproducible to 0.01"K. or to 1 part in 5000 in the vapor pressure. The first and third uncertainties displace the intercept but not the slope of the line describing the relationship between the cadmium activity and composition; only the second and fourth uncertainties are involved in the test of Raoult's law-the law predicts the slope of this act ivity-composition relationship while the intercept is set by the definition of activity.
Discussion of Results Figure 2, expansion A, shows our data at about 773OK. and those of Predel* at 801OK. In addition the solid curve aCd
=:
1 - 0.856NGa f 0.943N~a'
(3)
and the dashed curve aCd
=
+
1 - N G ~ 2.316Nca2
(4)
are plotted in both sections of the figure. The coefficient, 0.856, mas established from our data in the range 0.008 to 0.026 mole fraction cadmium, but a slope close to that value is also required if the Predel data are to be fitted with an additional b ' N B * term. Similarly with the b' term, 0.943, the value was established from the data of Predel, but it fits our data also. (An exception is made for the left-hand point in which the temperature controller had been functioning badly.) There appears to be a slightly greater curvature for our 755OK. data than for the curve tied to Predel's 801 OK. measurements. The sharper curvature is consistent with the change of activity with temperature; each group of Predel's measurements shows the same general form, but a t successively lower temperatures the departure from Raoult's law becomes slightly greater. I t may be noted that the coefficient, 2.316, fitted to Predel's data ai 0.892 mole fraction cadmium fits only that single poirt. Sormally it is expected that the b coefficient will lead to a fit of the data to perhaps 0.15 to 0.20 mole fraction of solute. S o other Raoult's law based b can be made to fit our data, either. This statement includes an equation for the cadmium free energy partial which can be derived approximately from Kleppa's equation (in kjoules)
AHM = 14.8Nc, - 2 5 N ~ , ~
(5)
937
for t'he mixing of gallium and cadmium at' the cadmiumrich end. The discrepancy cannot be resolved by invoking gas nonideality because the effect's of gas iniperfect'ion are almost totally cancelled'O in the activity calculation for tmhenearly pure solvent's vapor pressure. Since Raoult 's law has generally been considered to be a limiting law we must determine whet,her the gallium-cadmium data presented are in a concentration range which will effectively t'est the ey. 2 relationship. The data fitt'ed by the equation cover a concentration range froin about 0.75 to 0.992 mole fraction cadmium. Thus t'he average number of cadmium atoms separating gallium atoms ranges from about one in the more concentrated solutions to about five for the more dilute solution range. It' would seem significant that eq. 2 describes the data in this region in which the degree of interaction could be expected to undergo the greatest change. It would seem most unlikely that a significant deviation could occur at' a st'ill greater dilution. However, to evaluat,e effects at great dilution let us assume that the gallium activity coefficient for some unspecified cause should change even a niillionfold on going from 0.00001 mole fraction to infinite dilution. Following the usual Duheni predict'ion, the cadmium activity would change by A log YCd = 1/2(10-5)(6)= 0.00003 or Y C d would shift froin 1 t'o 1.00001. Then in the region where Henry's law was followed closely, the activity predicted for the solvent would follow a line almost paralleling Raoult's law but headed toward the point of zero cadmium activity and mole fraction. Thus the huge change assumed for the gallium behavior would cause only an experimentally indistinguishable deviation from the simple Raoult's law predict'ion! To correspond thermodynamically with eq. 2 behavior, the solute behavior would have to change violently and chemically unreasonably in the region of composition measured. Reliability of These Results. The agreement of our data with those of Predel is excellent and both groups of workers have used isopiestic techniques which can establish thermodynamic partial molal free energies very accurately. For example, t'he isopiestic balance (10) The Berthellot expression relating gas fugacity to actual and critical temperature and actual and critical pressures is
f
zz
P
+ (9T,PZ/128TP,)[1 - 6(Tc2TZ)]
For mercury11 T, = 1733'K. and P, = 1587 a t m . Using this equation a n d the mercury constants as an approximation t o the cadmium behavior in our system a t 0.974 mole fraction cadmium f C d alloyed/.fCd pure = ~ . o o o o o ~ ( ~s l lCo yde d / P C d p u r e )
This correction is very small and is in the wrong direction t o explain t h e experimental deviation from prediction. (11) F. Birch, Phys. Rea., 41, 641 (1952).
Volume 69, .Yumber 3
March 1966
938
G. R. B. ELLIOTT, J. F. LEMONS,
nieasures vapor activities to 1 part in 5000 or the free energy partial to about hO.3 cal. The precision is comparable with that of the very finest e.1ii.f. work, and the vapor pressures are unequivocally and reproducibly related to the values for pure
The Activity Relationships in Dilute Solutions It is possible to understand and, indeed, to anticipate the forms of the experimentally observed partial pressure relationships for solute and solvent in a dilute solution from the generally accepted view that such systems are composed of discrete niolecules (or atoms) mixed essentially randomly and interacting with each other through short-range12 forces. We will explore these relationships briefly. Special considerations peculiar to solutions of electrolytes will not be treated
The second factor, in simplified form, results froni AA, A-B, and B-B interactions which may be in the nature of bonding tendencies, repulsions, etc. The condensation rate depends 011 gas kinetics and on a gas condensation efficiency which approximates constancy in dilute solutions because the surface layer is coniposed predominantly of solvent (A) molecules. The familiar Henry’s law3 PB/PB‘
= ~ B N B
(64
relates the partial pressure and mole fraction of B, PB and N B , to the pure coniponent vapor pressure, PBO,by a unique constant, k g , characteristic of the systen1 under consideration. This law has experimental justificatio~i’~ and has logical ~ignificancel~ when it is realized that in sufficiently dilute solution each B niolecule is surrounded by A niolecules and is isolated from other €3 molecules essentially all of the time. Because all I3 niolecules which evaporate must break from similar environments ( 7 2 . , only A-B type interactions are involved) , the rate of escape from the surface is a linear function of (*oncentration. Since the rate of condensation is closely proportional to the vapor concentration, the pressure of B is correctly represented. When the voncentration of B is large enough to allow significant interaction between B niolecules, a single constant, 1r.B. can no longer reflect the activity-mole fraction ratio. The probability that a given molecule The Journal of l’hysical Chemistry
AND
H. S. SWOFFORD, JR.
of B will have another B molecule as a near neighbor, and thus have its tendency to escape altered, may be described by K N Bwhere K is the number of molecules in the shell around the B niolecule where the interactions become significant. The fraction of the total molecules thus changed will be ( K N B ) ( N B ) and the fraction unchanged will be (1-K N B()N B). Therefore the Henry’s law proportionality can now logically be expressed as the sum of two terms, one describing evaporation from a local environment comprised wholly of A molecules and the second involving the influence of another B molecule.
( l l a ) NOTEADDEDI N PROOF. A gross error in the A H of cadmium vaporization could also appear t o be a deviation from Raoult’s law. T h e equation from Kelley which we use is derived from A H V . = ~~~ 26,764 cal./mole. A value of 26,770 cal./mole is selected by R.Hultgren, R . L. Orr, P. D. Anderson, and K. K. Kelley in “Selected Values of Thermodynamic Properties of Metals and Alloys,” John Wiley and Sons, Inc., New York, N . Y., 1963. A 150 cal./mole uncertainty corresponds t o less than 3% of the deviation from Raoult’s law which we find. Recent work by Conant, Elliott, and Lemons with solutions of cadmium containing small amounts of nickel shows a distinctly different slope for the activity us. mole fraction curve from t h a t obtained for gallium-cadmium solutions. An apparent deviation from Raoult’s law due only t o a n error in A H would be expected t o result in t h e same slope for the two systems in the Raoult’s law region. We are indebted to Paul C. Nordine for pointing out t h a t the F d p terms for the effect of the different cadmium pressures upon t h e liquid were not discussed. These Vdp corrections would be very small and in the wrong direction t o explain t h e deviation. (12) Short-range forces, a s used here, means forces which are not significant a t more t h a n a few atoms distance. T h e argument is not altered, however, if longer range forces are considered so long a s t h e forces attenuate to insignifiance in some region. A dilute solution occurs when the force fields no longer overlap significantly. If thermodynamics is to have t h e extensive properties it requires for constant bulk properties, then the forces must be of shorter range than t h e smallest size specimen exhibiting the constant bulk property. (13) Most vapor pressure measurements have been made on concentrations great enough so t h a t a second term, a s in eq. 6c. must be included to describe accurate work. I n studies of t h e distribution of radioactive trace elements between two phases, t h e activity form of the relationship has received extensive justification. For example, separation plants for multigram amounts of plutonium were designed from studies a t concentrations sometimes only 10-lo as large. plus t h e assumption of Henry’s law. (14) See, for example, t h e discussion in G. N. Lewis, M . Randall, K. S. Pitaer, and L. Brewer, ”Thermodynamics,” McGraw-Hill Book Go., Inc., New York, N. Y., 1961, p. 239. This reference reaches different conclusions from those which will be reached here, however.
SOLVENT VAPORPRESSURES IN DILUTESOLUTIONS OF GALLIUM IN CADMIUM
every B molecule added has an influence on the A molecules surrounding it, this type of interaction must always be considered, too. Furthermore, it is obvious that if the solution is sufficiently dilute so that B-B interactions can be neglected, then each B molecule will influence an equal number of A molecules and the effect of the A-B type of interaction on the vapor pressure of A will be a linear function of the concentration of B. Similarly, if the sizes of B and A molecules are different, the volume fraction of A at the evaporating surface will be altered by an amount which in dilute solutions is a linear function of the concentration of B. In a very dilute solution the bulk of the A molecules are outside the sphere of influence of B molecules; consequently, their tendency to evaporate from solution will not differ froni that in pure solvent. Therefore for a solution, the unit activity of the pure solvent, A, must be altered by the sum of a number of factors, e.g., concentration, relative volume, and various A-B interactions, all of which are proportional to the concentration of B. These proportionality constants may be added to give a single constant k~ so that
The probability that a molecule of A already having a single molecule of B in its environment will have a second molecule of B as a near neighbor is again proportional to the concentration of B. The treatment is similar to, but more complex than, that described for the solute. Therefore when the concentration of B is sufficiently large a second term must be included which sums all of these possible effects due to two molecules of B influencing the environment of a single A molecule. The resulting equation can be reduced to eq. 2. Although eq. 6c and 2 show an obvious similarity in form, the respective constants are different and would not be expected to bear a simple relationship to each other. They reflect related interactions, but the effect of the interactions on the evaporation processes is complex. Raoult’s Law. If the partial pressure of the solvent is to be predicted from the partial pressure of the solute, then the function relating them must take into account all types of interaction which affect the pressure, ;.e., A-A and A-B types of interaction for very dilute solutions. For tht: dilute solution region in which Henry’s
939
law is observed to hold it is conventional to make use of the Gibbs-Duhem relationship in conjunction with Henry’s law to arrive a t the Raoult’s law relationship16
PA/PAO = N A F 1 -
(8)
In those concentration ranges in which B-B type interactions must be considered this equation can be modified in a manner already described to give eq. l . l 5 However, the partial molal free energy as conventionally formulated and employed to arrive at eq. 7 suffers because it is unable to make use of the A-B type of interaction contained in Henry’s law. This difficulty is most apparent when it is seen that for the usual derivation it is necessary to use the derivative of the partial molal free energy and that this derivative as formulated is independent of the Henry’s law constant, which disappears from the relationship. 5’ince this Henry’s law constant is our only measure of A-B interaction and since A-B interactions are an essential factor in determining the partial pressure of we have lost a vital element in the relationship. This difficulty is further emphasized when the relationship is put in activity form15 k
(d In YA/dNA)
= - (NB/NA)(d
In rB/dNA)
(9)
Since in the Henry’s law concentration region k B is constant and equivalent to Y B , eq. 9 requires that y A be invariant over the same concentration range. Thus eq. 8 and 9 require that the vapor pressure of the solvent be a function of concentration alone for every solution regardless of the size or chemical character of the actual chemical species present in the solution. Although the fact that the type of solvent molecule clearly influences the partial pressure of the solute as shown by Henry’s law, these equations assert that these same A-B interactions are without effect on the partial pressure of the solvent. This has been shown to be contrary to fact for the gallium-cadmium system.
Acknowledgment. We wish to acknowledge contributions by Dr. Donald R. Conant to the views expressed in this paper. A more thorough presentation of these views will be given in a paper to be published shortly.16 (15) L. S. Darken and R. W.Gurry, “Physical Chemistry of Metals.” hlcGraw-Hill Book Co., New York, K. Y.,1953, pp. 259-261 (also numerous other textbooks.) (16) D. R. Conant, Los Alamos Scientific Laboratory Report. in preparation.
Volume 60,Sumber 3
March 1966
